Stability Patterns in the Homology of Moduli Spaces

同源性是一百多年前引入的一种数学工具，用于测量在变形下保持不变的形状特征。这有助于在微积分和物理学中进行严格的计算，这些计算涉及在有“洞”的空间上定义的函数的积分。同调给出了几何对象中孔洞的代数度量，允许代数技术给出几何信息，例如，表明一个给定的几何对象不能变形成另一个。本课题的目的是研究数论和几何空间族同调中的模式。这些模式通过将无限计算减少到有限计算来简化同调计算。在这个项目中，将开发新的方法，包括在理论代数拓扑和数论中使用计算机。该项目将通过指导、研讨会和会议加强代数拓扑的研究生和博士后培训。该项目将通过参加一个夏季桥梁项目来促进多样性和包容性，该项目是为来自未被充分代表的种族和族裔群体的学生设计的，旨在为他们进入研究生院做更好的准备。本项目旨在提高对算术群同调的理解，同调是数论、代数k理论甚至流形理论中的一个核心概念。在高维中，已知同调会消失，而在低维中，已知同调会稳定。在许多情况下，这些稳定的同调群已经完全计算出来了。该项目侧重于两个维度范围：刚好低于已知同源性消失的范围，刚好高于稳定范围。据推测，最高程度的同源群应该表现出一种称为“极端稳定性”的模式，而接近稳定范围的同源群应该表现出一种称为“二级稳定性”的不同模式。高度连接的简单复合体和操作细胞将被用来试图建立这些猜想。在研究算术群的同调时，与代数k理论的联系将是一个重点。类似的模式将研究来自几何拓扑的空间，如各种模空间和组态空间。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Homology is a mathematical tool that was introduced over a hundred years ago to measure features of shape that remain invariant under deformation. This helped make rigorous many calculations in calculus and physics involving integrals of functions defined on spaces with “holes.” Homology gives an algebraic measure of holes in geometric objects, allowing algebraic techniques to give geometric information, for example, showing that a given geometric object cannot be deformed into another one. The aim of this project is to study patterns in the homology of families of spaces coming from number theory and geometry. These patterns simplify homology calculations by reducing infinite calculations to finite calculations. In this project, new methods involving the use of computers in theoretical algebraic topology and number theory will be developed. The project will enhance graduate and postdoctoral training in algebraic topology through mentoring, seminars, and conferences. The PI will promote diversity and inclusion through participation in a summer bridge program designed for students from underrepresented racial and ethnic groups to better prepare them for graduate school.This project aims to improve understanding of the homology of arithmetic groups, a central concept in number theory, algebraic K-theory, and even the theory of manifolds. In high dimensions, the homology is known to vanish, and, in low dimensions, the homology is known to stabilize. These stable homology groups have been completely calculated in many cases. The project focuses on two ranges of dimensions: just below where the homology is known to vanish and just above the stable range. Conjecturally, the highest degree homology groups should exhibit a pattern called “extremal stability” and the homology near the stable range should exhibit a different pattern called “secondary stability.” Highly connected simplicial complexes and operadic cells will be used to try to establish these conjectures. The connection with algebraic K-theory will be a key point of emphasis when studying the homology of arithmetic groups. Similar patterns will be investigated in spaces coming from geometric topology such as various moduli spaces and configuration spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.